This document outlines how multi-agent systems can support distributed graph transformations. It discusses graph grammars and their complexity, and how an agent system can facilitate parallel graph computations by decomposing graphs into complementary subgraphs distributed among agents. A case study demonstrates how agents coordinate to apply graph productions and incorporate vertices while avoiding conflicts.

1. DISTRIBUTED GRAPH
TRANSFORMATIONS
SUPPORTED BY MULTI-
AGENT SYSTEMS
Adam Sędziwy, PhD
AGH University of Science and Technology
Cracow, Poland

2. 1. Objectives
2. Graph transformations. The complexity
3. Faciliating graph transformations
4. The role of an agent system
5. Case study
6. Examples of applications
7. Summary
OUTLINE

3. Graph models are widely used for describing a range of
systems. For that reason we focus on:
 Developing the effective method of graph-based computations
 Computations have to be performed in parallel
 Computations are intended to be carried out by an agent
system
OBJECTIVES

4.  A graph grammar is a generalization of a string grammar
notion:
 The example:
 Terminal and nonterminal symbols are replaced by terminal
and nonterminal vertices (context-free grammar) or graphs
(context grammar)
GRAPH GRAMMARS

5.  The sample graph production: 𝑃: 𝑋 → 𝑅
 Q: how to embed R in G-X ? A: Embedding rule is associated with
each production.
1. Edges {i,X} where i≠a , should be removed
2. The (former) edge {a,X} should be replaced with {a,Y} and {a,e}
GRAPH GRAMMARS (CONT.)

6.  The basic problem: exponential complexity of parsing and
membership problem
 A graph grammar's parsing complexity vs expressiveness
 If some restrictions are imposed on a grammar then the
complexity reduction is possible (edNLC, ETPL(k), … )
COMPLEXITY ISSUES

7.  A graph should be decomposed into complementary
subgraphs.
– Q: How?
– A: It depends on a problem: equally sized subgraphs, minimized
number of connections, minimized redundancy,...
 Processing is performed in parallel with some data
exchange.
 The desirable property: centralized graph grammar rules
apply to be used in a distributed environment
FACILITATING GRAPH COMPUTATIONS –
"WHAT IS TO BE DONE”?

8.  Decomposition based on a vertex replication
GRAPH DECOMPOSITION –
VERTEX REPLICATION

9.  Motivation: applicable to subproblems
requiring an autonomous activity
(decision making) rather than executing
a deterministic algorithm
 Particular subtasks (e.g., local graph
transformations) are performed by agents
 An agent's knowledge is represented by a
graph structure (data) and a grammar
(action)
 Agents have to cooperate to achieve their
goals
AGENT SYSTEM ON
A DISTRIBUTED GRAPH

10.  Centralized graph decomposition
 The scenario of a local production
 A node’s incorporation
 Conflicts
CASE STUDY

11.  The graph G in a centralized and ditributed form
CASE STUDY - DECOMPOSITION

12.  The agent A1 aims at applying the production P to G1
 A1 has to obtain an exclusive access to the vertex v
CASE STUDY – THE LOCAL PRODUCTION

13.  Replicas of v have to be removed from other subgraphs 
incorporation of the vertex v to G1
 To preserve the overall consistency
the agent A1 requests other agents
to lock relevant vertices (e.g., u,w)
while incoroprating v
 The source of conflicts!
CASE STUDY - INCORPORATION

14.  A1 determines a set HA (L) of other agents hosting vertices of L
 A1 follows 2PC:
PHASE 1:
 A1 requests agents Ai HA(L) to block vertices of L
 Ai responds with AGREEMENT=YES | NO:BLOCKED | NO:NONEXIST
PHASE 2:
 [AGREEMENT=YES] A1 sends COMMIT; Ai responds with a relevant part
of L
 [AGREEMENT=NO:BLOCKED] A1 sends ABORT to HA(L) and restarts
with a random delay
 [AGREEMENT=NO:NONEXIST] A1 sends ABORT to HA(L) and tries to
determine HA(L) again
CASE STUDY – INCOROPRATION (CONT.)

15. CASE STUDY – INCOROPRATION (CONT.)
 The agent A1 has incorporated the vertex v

16.  Embedding rules:
 {p,q} is replaced by {p,a}
 {p,r} is replaced by {p,b}
 {u,v} is replaced by {u,a}
 {w,v} is replaced by {w,b}
 {q,v}, {r,v} are removed
CASE STUDY - PRODUCTION
P

17.  A border node v may be shared by all complementary graphs
(pessimistic case), i.e., O(N).
 Pessimistically, the number of vertices to be locked is:
𝑛 𝐿 = 𝑂(𝑁 ∗ 𝑑 𝑚𝑎𝑥)
 The additional issue: a number of exchanged messages
 Solution: the another approach to the graph decomposition –
the graph slashing
CONFLICTS

18.  Slashed graphs concept. A single edge is shared by exactly
two agents
GRAPH DECOMPOSITION –
SLASHING
𝑛 𝐿 = 𝑂(𝑑 𝑚𝑎𝑥)

19.  CAD system for the architectural design
 Separate design processes and graph models for the interior
and exterior
 Both processes may clash on shared elements (e.g., windows):
coordination required
EXAMPLE 1

20.  The control in smart lighting systems
 The lighting control is based on sensor data (reflecting an
environment state)
 Lamps’ performance is controlled by relevant graph grammar
productions triggered by changes in an environment
EXAMPLE 2

21.  The graph-based representation (GBR) of systems is the
suitable formal approach to model a range of systems.
 GBR may be easily decomposed becoming an environment for
an agent system deployment
 A decomposition method depends on a problem. Following
factors should be considered:
 Statistical properties of an agent system’s behavior (avoiding
conflicts)
 A number of messages required for completing atomic operations
SUMMARY

22. 1. L. Kotulski, A. Sędziwy, B. Strug: Heterogeneous graph
grammars synchronization in CAD systems supported by
hypergraph representations of buildings, Expert Systems
with Applications, Elsevier,
http://dx.doi.org/10.1016/j.eswa.2013.07.043
2. I. Wojnicki, S. Ernst, L. Kotulski, A. Sędziwy: Advanced
Street Lighting Control, Expert Systems with Applications,
Elsevier, http://dx.doi.org/10.1016/j.eswa.2013.07.044
3. A. Sędziwy: Effective Graph Representation for Agent-Based
Distributed Computing, Lecture Notes in Computer Science,
Vol. 7327, pp 638-647, Springer, 2012
4. L. Kotulski, A. Sędziwy: Parallel graph transformations
supported by replicated complementary graphs, Lecture
Notes in Computer Science, Vol. 6594, pp 254-264,
Springer, 2011
FOR FURTHER READING

23. THANK YOU FOR YOUR ATTENTION