** **

**GLOSSARY**

**Adjacency **is the graph theoretic
expression of the fact that two entities, represented by nodes, are directly
related, tied, or connected with one another. Formally, given entities
*n*_{i} and *n*_{j }in a set of agents
** N**, and the

**a density Criterion: **a procedure for tie assignment in
the construction of a blockmodel. According to
this criterion, an arc (**b**_{AB}) between two blocks (A,B) for a
given relation is 1 if the observed density of arcs between the two blocks
(*d*_{AB}) is at least as large as **a**, and zero otherwise.
Often, a is chosen to equal
(** D**), the density of
the original sociomatrix. Formally, for

**arc:** in graph theory, an arc is a
directional relation that has an explicit origin
and destination, represented by a line between entities with an arrowhead at
the destination.

**backward linkage **= transaction link(s) *to* the buying sector of
interest *from* industries that supply intermediate goods or services. The
supplying industries are also known as 'upstream' sectors in the flow of goods
from extraction through raw, processed, and final retailed form to consumers.
The magnitude of a sector's backward linkages are given by its column
coefficients in the multiplier matrix.

**betweenness,** a measure of the *global centrality* of the actor
i, is the probability that the shortest path from
actor j to actor k takes a route through agent i. Formally,
** B**(

**blockmodel **is an aggregated sociomatrix representation of a
network. A blockmodel is created from an elemental sociomatrix data base by (i)
partitioning the entities into discrete subsets, called "blocking," and (ii)
assigning 0/1 ties between each pair of blocked subsets (White, Boorman and Breiger, 1976; Holland and Leinhardt,
1979).

**closeness** is a second measure of the *global centrality* of
the actor i. ** C**(

**complement**: generally, an input or consumer good whose use
decreases as the cost of another rises is considered a *complement in
demand* to that other good. An output or product whose supply decreases as
the price of another falls is considered a *complement in supply *to the
other product. Complements generally occur in fixed proportions in an economy.
For **Keystone Sector Analysis:** *entities I and J are perfect
complements when K's interaction with I is always accompanied by K's
interaction with J* (Kilkenny and Nalbarte-this
paper).

A **complete** graph is one in which all the actors have two-way ties
to all other actors.

A **component**
is the largest subset of actors in a network that all relate to each
other, also known as *group *or a *sub-graph.* A **strong
component** is one in which the arcs that make up the paths are aligned in continuous chain without a change
of direction. A **weak component** is made of actors that are linked by
non-directional edges (Scott, 1991).

**cut-point** is the node whose removal from the system would
increase the number of components by dividing the
graph into 2 or more separate components, between which there are no
ties (Scott,
1991).

**density**
is the measure of how many entities are related to others in a set. Density
(** D**) is measured by the ratio of the actual number of
non-reflexive arcs in proportion to the maximum
possible number of non-reflexive arcs:

**dichotomous relation** is a tie between two
agents (dyad) that either does or does not exist. It is recorded as a binary
variable with (1) indicating the presence and (0) indicating the absence of the
relation between the two set entities in the dyad.

**directed**
**graphs **or** digraphs **are the graphic representations of
directional relational data among entities in a
set. Entities are illustrated as nodes, and the directional relations, if they
exist, are illustrated as arcs, with the arrowhead
pointing from the source or sending node to the destination or receiving node.
Formally, a **digraph** is a finite, non-empty set ** N**,
whose elements

**directional relation** is an interaction between entities that is specific
about which entity is the source versus the sink, which is the origin or
destination. It is represented graphically by an arc, a line between entities with an arrowhead at the
destination. The sociomatrix for a directional
relation will not generally be symmetric, unless all
ties are reciprocated.

**dominant industry**: the sector(s) with relatively high
location quotients, that also have a number of
input-output linkages with other local industries with high location quotients
Cella 1984

A **dyad**
consists of a pair of actors and the possible ties
between them (Wasserman and Faust, 1994).

**edge**:
in graph theory, this represents a non-directional
relation or tie which
is non-specific about the origin or destination of the flow on the link. It is
illustrated by a line between the interacting agents that has no arrowhead.

**efficient path test** is applied to a blockmodel
of a community to determine if a sector is a keystone. If the excision of the sector from the
image matrix (or reduced
graph) reduces the **closeness** measures for the remaining sectors, that sector is a
**keystone**.
(Kilkenny and Nalbarte, this paper)

**forward linkage **the transaction link(s) *from* the supplying sector
of interest *to *industries that demand intermediate goods or services.
The using industries are also known as 'downstream' sectors in the flow of
goods from extraction through raw, processed, and final retailed form to
consumers. The magnitude of a sector's forward linkages are given by its row
coefficients in the multiplier matrix.

**fracture test** is a cut-point test based
on ties applied to a blockmodel of a community to determine if a sector is a
keystone. If the excision of the sector from the
image matrix (or reduced
graph) destroys the connectivity of the network by increasing the number
of **components, **that sector is a **keystone**. (Kilkenny and Nalbarte, this paper)

**geodesic** *d*(*n*_{i},*n*_{j}) is
the shortest path between two nodes i and j. It is measured as the number of
arcs required to get from i to j, which is the first power *p* for which
the ijth element of *A*^{p}^{} is non-zero:
*d*(*n*_{i},*n*_{j}) = min{*p*}|
*A*_{ij}^{P }^{}>0 (Wasserman and Faust, 1994). Equivalently, for any
digraph with adjacency matrix ** A, **each cell

**image matrix** is the aggregated form of an elemental
sociomatrix, representing the
blockmodel in a
matrix comprised of [0,1] cell entries.

**immiserizing growth** is expansion in a region's real economic output that
earns less rather than more real income. This outcome can arise only under two
conditions (1) the region must be large in the market for the product, where
'large' means that changes in the region's supply of the product will affect
market prices, and (2) demand for the product must be highly inelastic, such
that revenues fall even as more is sold.

**indegree** measures the strength of a node as a sink in a system.
It is the number of arcs ending at the node,
measured by the column sum for the node in a dichotomous
sociomatrix: formally the indegree of actor
j = å_{i} *a*_{ij} .

**keystone sector** is the type of entity (business, institution,
organization, etc) in a community that plays a unique role and without which
the community is fundamentally and detrimentally altered (Kilkenny, 1997). Based on the terms
Keystone Species and turnkey
or key sector.

**keystone species** is the species
responsible for the structure and integrity of an ecosystem. The term was first
coined by ecologists in the late 1960s (Paine,
1969).

**local centrality** is a measure of prominence which reflects the number of
direct transmissions from the entity, measured by the outdegree (or row sum) for the entity. Also known as
*degree centrality.*

**local prestige** is a measure of prominence which reflects the number of
the entity's direct receipts, measured by the indegree
(or column sum) of the entity. Also known as *degree prestige.*
(Wasserman and Faust, 1994).

**location quotient: **a measure of relative concentration in an area. Given a
local proportion, such as the percentage of total region r employment in
industry i (e_{ir}), and a reference proportion, such as the share of
nationwide employment in industry i (e_{ir}), the location quotient
(LQ_{ir}) is e_{ir}/e_{i.}, more formally:
LQ_{ir} = (e_{ir}/å _{i
}e_{ir})/( å_{r}e_{ir}/å_{i}å_{r}e_{ir}) .

**multiplier matrix** is an array of numbers which show the amount change in
the row sector due to a unit change in the column sector. It is calculated from
an industry-by-industry transactions matrix in three steps as follows. First,
divide the transaction cells by their column totals. This gives a matrix of
sectoral expenditure shares [A]. Second, subtract this matrix from the Identity
matrix [I-A]. Third, invert this subtrahend. The result is the multiplier
matrix "*m*". Formally, *m *= [I-A]^{-1} .

**nodes**
represent the individual entities or actors in networks.

**non-directional relation** is an interaction between entities that is non-specific
about which entity is the source versus the sink, or which is the origin or
destination. It is represented graphically by an edge or tie, a simple line
between the entities, and by a symmetric sociomatrix.

**Oneblock Criterion**: a procedure for tie assignment in the construction of
a blockmodel. According to this criterion, an arc
(**b**_{AB}) between two blocks (A,B) for a given relation is 1 only
if all possible arcs (*t*_{ij}) from all actors in
the row block to actors in the column block exist, otherwise the block arc is
0. Formally, **b**_{AB} = 1 if *t*_{ij}
=1 for all i Î A and all j Î B ; else **b**_{AB} = 0.

**Outdegree **measures the strength of a
node as a source in a system. It is the number of arcs beginning at a node,
measured in dichotomous sociomatrix data as the row sum for the node:
outdegree of actor i = å_{j} *a*_{ij}

**reduced digraph** the illustration of arcs
between the blocks of nodes for a blockmodel. See
also image matrix.

**reflexive tie** is the relation that a
particular entity has with itself. In a sociomatrix, a reflexive tie by the ith
entity is recorded by *a*_{ii} = 1. In a digraph, reflexive ties
are drawn as arrows that originate and end on the same node, that are curved
back on themselves.

A **relation **is the collection of ties of a specific kind among a set
of entities**. **Alternatively, consider the mathematical* *definition
of **binary relation **(Robinson and
Foulds,1980):

Given two setsSandT, each member of setSmay be related to a number (perhaps zero) of members of setT. The mathematical description of this situation is called a binary relation. IfsSandtT, then (s,t) is a member of this set whensis related tot.

**Sinks Substitute Criterion: **a
procedure for tie assignment in the construction of a blockmodel. According to this criterion, an arc
(**b**_{AB}) between two blocks (A,B) for a given relation is 1 if
there is an arc (*t*_{ij}) from every actor in the
row block to at least one actor in the column block, otherwise the block tie is
0. Formally, (Kilkenny and Nalbarte, this paper)

**social embeddedness** in traditional societies,
economic life is submerged ('embedded') in social relations. For example,
businesses employ locals and buy locally, no matter what, because it is
expected of them by society. In contrast in modern life, social relations are
often an epiphenomenon of the market (Granovetter, 1985). For example,
when businesses employ locals or buy locally simply because it is profitable,
this is the reverse of *social embeddedness*.

A **social network** consists of a finite set of *actors* and the
*relation* or relations defined on them. **Actors** are social
entities, discrete individuals, corporate or collective social units. (Wasserman and Faust, 1994).

**sociomatrix**
is an entity-by-entity array of data on the relational ties between them. Rows of the sociomatrix
represent the sending actors while the columns represent the receiving actors.
See also digraph.

**Sources Substitute Criterion: **a
procedure for tie assignment in the construction of a blockmodel. According to this criterion, an arc
(**b**_{AB}) between two blocks (A,B) for a given relation is 1 if
there is an arc (*t*_{ij}) to every actor in the
column block from at least one actor in the column block, otherwise the tie is
0. Formally: .(Kilkenny and Nalbarte, this paper)

**structurally equivalent** entities have exactly the same directional ties (arcs) to and from all other entities with whom either
has ties. Formally, entities i and j are *structurally equivalent *if, for
all other actors k = 1,2,...** N**; k¹i,j, and all the relations r = 1,2,...,

**substitute**: generally, an input or consumer good whose use
increases as the cost of an alternative rises is considered a *substitute in
demand* for that alternative. An output or product whose supply increases as
the price of an alternative falls is considered a *substitute in supply
*to the alternative product. Substitutes generally satisfy similar demands
(play similar roles) in an economy. For **Keystone Sector Analysis:**
*entities I and J are perfect substitutes if a slight decrease in the
desirability of K interacting with I leads to K interacting with J instead
(*Kilkenny and Nalbarte, this
paper*).*

**tie** is
a relation between two entities in a network. If it has direction it is an arc, if
non-directional it is an edge.

**transitive **a relation R is *transitive* over the set
{**X**} if for all x_{i,j or k} in {X}, if x_{i} R x_{j
}AND x_{j} R x_{k}, then x_{i} R x_{k}.
For example, let R be "like" and {X} be people. Like is a transitive relation
if every time a person A likes B, and person B likes C, then person A likes C.
It is easy to find a counterexample to prove that in fact, "like" is** not**
a transitive relation for people.

**turnkey or key sector **sectors whose structure of backward and forward
linkages create above-average impacts on the rest of the economy.
Activities having the highest linkages are considered key
sectors because it is thought that concentrating resources in them will
stimulate more production, income and employment than alternative allocations
of resources. (Cella, 1984). See also** **dominant industry.

**valued relation** is a
tie that has a magnitude that reflects the level,
intensity, or frequency of a relation.

**Zeroblock Criterion: **a procedure for
tie assignment in the construction of a blockmodel. According to this criterion, a tie
(**b**_{AB}) is assigned between two blocks (A,B) for a given
relation is 0 only if there are no arcs (*t*_{ij} =
0) from any actor (i) in the row block to any actor (j) in the column block;
otherwise the block arc is 1. Formally, **b**_{AB} = 0 if
*t*_{ij} =0 for all i Î A and all j Î
B else **b**_{AB} = 1.