Multilayer Networks 101

This guide explains what multilayer networks are and why they matter for modern network analysis.

What are Multilayer Networks?

A multilayer network is a complex network structure that goes beyond traditional single-layer graphs by incorporating multiple types of relationships, node types, or interaction contexts. They model the reality that most real-world systems involve multiple, interconnected types of relationships.

Traditional vs. Multilayer

Traditional (Single-Layer) Networks:

One type of node (e.g., only people)
One type of edge (e.g., only friendship)
Homogeneous structure
Limited ability to model complex systems

Multilayer Networks:

Multiple node types (e.g., people, organizations, documents)
Multiple edge types (e.g., friendship, collaboration, citation)
Multiple layers of interaction
Inter-layer and intra-layer connections
Rich, heterogeneous structure

Types of Multilayer Networks

py3plex supports several common multilayer network paradigms:

Multiplex Networks

Definition: Multiple layers with the same set of nodes but different types of edges.

Characteristics:

Node set is identical across layers
Each layer represents a different relationship type
Inter-layer edges typically connect the same node across layers

Examples:

Social networks: The same people connected via friendship, colleague, and family relationships
Transportation: Cities connected via air, rail, and road networks
Communication: Users interacting via email, phone, and instant messaging

Visual representation:

Layer 1 (Friends):        Layer 2 (Colleagues):
A --- B --- C             A --- B
  \   |                       |
    \ |                       D
      D

Code example:

from py3plex.core import multinet

network = multinet.multi_layer_network(network_type="multiplex")

# Same nodes, different relationship types
network.add_edges([
    ['Alice', 'friends', 'Bob', 'friends', 1],
    ['Alice', 'colleagues', 'Bob', 'colleagues', 1],
], input_type="list")

Heterogeneous Networks (HINs)

Definition: Networks with different node types and type-specific relationships.

Characteristics:

Multiple node types (e.g., authors, papers, venues)
Edges connect nodes of specific types
Meta-paths describe relationship sequences

Examples:

Academic networks: Authors write papers published in venues
E-commerce: Users purchase products from sellers
Biomedical: Drugs treat diseases via molecular targets

Visual representation:

Authors       Papers        Venues
[Alice] ----> [P1] --------> [ICML]
   |           ^               ^
   |           |               |
   +---------> [P2] -----------+
[Bob]

Code example:

network = multinet.multi_layer_network()

# Different node types on different layers
network.add_edges([
    ['Alice', 'authors', 'Paper1', 'papers', 1],
    ['Paper1', 'papers', 'ICML', 'venues', 1],
], input_type="list")

Temporal Networks

Definition: Networks that evolve over time, with time-sliced layers.

Characteristics:

Each layer represents a time period
Nodes may appear/disappear over time
Edges show relationships at specific times
Inter-layer edges connect temporal states

Examples:

Communication networks: Who-contacts-whom over different time periods
Social dynamics: Friendship evolution over years
Disease spread: Contact networks during epidemic progression

Visual representation:

t=1: A --- B --- C
          |
          D

t=2: A --- B --- C --- E
          |       |
          D ------+

t=3: A --- B --- C --- E
                 |
                 D

Code example:

network = multinet.multi_layer_network()

# Different time periods as layers
network.add_edges([
    ['A', 't1', 'B', 't1', 1],
    ['A', 't2', 'B', 't2', 1],
    ['B', 't2', 'D', 't2', 1],
], input_type="list")

Interdependent Networks

Definition: Multiple networks where nodes in one network depend on nodes in another.

Characteristics:

Networks with distinct functions
Dependencies between networks
Cascading failures possible
Critical infrastructure modeling

Examples:

Infrastructure: Power grid depends on communication network
Supply chain: Manufacturing depends on logistics network
Cyber-physical systems: Software layer depends on hardware layer

Visual representation:

Power Grid:           Communication Net:
P1 --- P2 --- P3      C1 --- C2 --- C3
 |      |              |      |
Depends on            C4      |
 |      |                     |
 v      v              v      v
 C1 --- C2            P1 --- P2

When to Use Multilayer Networks

Use multilayer networks when:

1. Multiple Relationship Types Matter

If your system has multiple types of connections that interact, a multilayer model is essential.

Example: Studying information diffusion in social networks where both online and offline relationships matter.

2. Node Roles Vary by Context

When nodes play different roles in different contexts or layers.

Example: A person might be central in their work network but peripheral in their hobby network.

3. Layer Interactions Are Important

When understanding how layers interact is crucial to your analysis.

Example: How transportation failures in one mode (air) affect alternatives (rail, road).

4. Temporal Evolution Matters

When the timing and sequence of relationships are important.

Example: Understanding how community structure evolves over time in a social network.

5. System-Level Properties Emerge

When whole-system properties can’t be understood by analyzing layers independently.

Example: Resilience of infrastructure systems depends on cross-layer dependencies.

Comparison with Single-Layer Approaches

Why Not Just Aggregate?

You might ask: “Why not just combine all layers into one network?”

Problems with aggregation:

Loss of Information: Different relationship types get conflated
Misleading Metrics: Centrality computed on aggregated network can be wrong
Hidden Patterns: Layer-specific patterns are lost
Incorrect Analysis: Communities and pathways depend on layer structure

Example:

# DON'T: Naive aggregation loses information
all_edges = []
all_edges.extend(friends_edges)
all_edges.extend(colleague_edges)
single_layer = nx.Graph(all_edges)  # Information about edge types lost!

# DO: Use multilayer representation
network = multinet.multi_layer_network()
network.add_edges(friends_edges + colleague_edges, input_type="list")

Why Not Analyze Each Layer Separately?

You might ask: “Why not just analyze each layer independently?”

Problems with separate analysis:

Ignores Cross-Layer Effects: Nodes may be important because of multi-layer presence
Misses Inter-Layer Patterns: Communities often span multiple layers
Incomplete View: Node importance depends on cross-layer activity
Lost Synergies: Complementary information across layers is ignored

Example: A person who is moderately connected in multiple social contexts may be more influential than someone highly connected in just one context.

Key Concepts

Intra-Layer Edges

Edges within a single layer connecting nodes in that layer.

Example: Friendships within the “friends” layer.

Inter-Layer Edges

Edges between layers, typically connecting the same node across layers or different nodes in different layers.

Example: Connecting Alice in the “friends” layer to Alice in the “colleagues” layer.

Node-Layer Pairs

In py3plex, nodes are represented as (node_id, layer_id) tuples.

Example: ('Alice', 'friends') and ('Alice', 'colleagues') are different node-layer pairs.

Supra-Adjacency Matrix

A matrix representation that stacks layer adjacency matrices into a block structure, encoding both intra-layer and inter-layer connections.

See py3plex Core Model for implementation details.

Real-World Applications

py3plex has been used for analyzing:

Biological Networks:

Protein-protein interactions with multiple evidence types
Gene regulatory networks with transcription, translation, and metabolic layers
Brain networks with structural and functional connectivity

Social Networks:

Multi-platform social media analysis (Twitter + Facebook + Instagram)
Relationship type analysis (friends, family, colleagues)
Online-offline integration studies

Citation Networks:

Author-paper-venue multilayer structures
Knowledge graphs with entities, relationships, and contexts
Research collaboration networks

Transportation:

Multi-modal networks (bus, train, metro, air)
Urban mobility analysis
Resilience and failure analysis

Infrastructure:

Critical infrastructure interdependencies
Smart city systems
Cyber-physical systems

Multilayer Networks 101

What are Multilayer Networks?

Traditional vs. Multilayer

Types of Multilayer Networks

Multiplex Networks

Heterogeneous Networks (HINs)

Temporal Networks

Interdependent Networks

When to Use Multilayer Networks

Comparison with Single-Layer Approaches

Why Not Just Aggregate?

Why Not Analyze Each Layer Separately?

Key Concepts

Intra-Layer Edges

Inter-Layer Edges

Node-Layer Pairs

Supra-Adjacency Matrix

Real-World Applications

Further Reading